Section 8-1: Geometric Means SOL: None Objective: Find the geometric mean between two numbers Solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse Vocabulary: Geometric Mean between two numbers is the positive square root of their product Arithmetic Mean between two numbers is their average Key Concept: Geometric mean: for any positive numbers a and b, the geometric mean is positive, a --- = --- or with cross products 2 = ab or = ab b Application of Geometric Mean a hypotenuse (length = a + b) altitude b From similar triangles a --- = --- b Geometric mean of two numbers a, b is square root of their product, ab The length of an altitude,, from the 90 angle to the hypotenuse is the geometric mean of the divided hypotenuse = ab Theorems: Theorem 8.1: If the altitude is drawn from the verte of the right angle of a right triangle to its hypotenuse, then the two triangles formed are similar to the given triangle and to each other. Theorem 8.2: The measure of an altitude drawn from the verte of the right angle of a right triangle to its hypotenuse is the geometric mean between the measures of the two segments of the hypotenuse. Theorem 8.3: If the altitude is drawn from the verte of the right angle of a right triangle to its hypotenuse, then the measure of a leg of the triangle is the geometric mean between the measures of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
Eample 1: Find the following: Arithmetic Mean (AM): is the average of two numbers eample: 4 and 10 Geometric Mean (GM): is the square root of their product eample: 4 and 10 Eample 2: Find the AM and GM of the following numbers: AM GM a. 6 and 16 b. 4 and 8 c. 5 and 10 d. 2 and 14 Eample 3: Find the length,, of the altitude: Eample 4: In ABC, BD=6 and AD=17, find CD Eample 5: Find c and d in JKL Concept Summary: The geometric mean of two numbers is the square root of their product You can use the geometric mean to find the altitude of a right triangle Homework: pg 535-37; 1-3, 5, 8, 9, 18, 36
Section 8-2: The Pythagorean Theorem and its Converse SOL: G.7 The student will solve practical problems involving right triangles by using the Pythagorean Theorem, properties of special right triangles, and right triangle trigonometry. Solutions will be epressed in radical form or as decimal approimations. Objective: Use the Pythagorean Theorem Use the converse of the Pythagorean Theorem Vocabulary: None new Theorems: Theorem 8.4: Pythagorean Theorem: In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse: a 2 + b 2 = c 2 Theorem 8.5: Converse of the Pythagorean Theorem: If the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle Pythagorean Theorem a c Pythagorean Theorem a 2 + b 2 = c 2 Sum of the squares of the sides is equal to the square of the hypotenuse b Eample 1: Find QR in the figure to the right Eample 2: Find BC in the figure to the left
Key Concept: Pythagorean Triple: Three whole numbers that satisfy a 2 + b 2 = c 2 Common eamples: 3,4,5; 5,12,13; 6,8,10; 7,24,25, 8,15,17; 9,12,15; 9,40,41; Eample 3: Determine whether 9, 12, and 15 are the sides of a right triangle. Then state whether they form a Pythagorean triple. Eample 4: Determine whether 21, 42, and 54 are the sides of a right triangle. Then state whether they form a Pythagorean triple. Eample 5: Determine whether 4 3, 4, and 8 are the sides of a right triangle. Then state whether they form a Pythagorean triple. Eample 6: Determine whether each set of measures are the sides of a right triangle. Then state whether they form a Pythagorean triple. a. 6, 8, 10 b. 5, 8, 9 c. 3, 5, 14 Concept Summary: The Pythagorean Theorem can be used to find the measures of the sides of a right triangle If the measures of the sides of a triangle form a Pythagorean triple, then the triangle is a right triangle Homework: pg 545-47; 1-3, 9, 10, 18, 21-23, 27
Section 8-3: Special Right Triangles SOL: G.7 The student will solve practical problems involving right triangles by using the Pythagorean Theorem, properties of special right triangles, and right triangle trigonometry. Solutions will be epressed in radical form or as decimal approimations. Objective: Use properties of 45-45 -90 triangles Use properties of 30-60 -90 triangles Vocabulary: None new Theorems: Theorem 8.6: In a 45-45 -90 triangle, the length of the hypotenuse is 2 times the length of the leg. Theorem 8.7: In a 30-60 -90 triangle, the length of the hypotenuse is twice the length of the shortest leg, and the length of the longer leg is 3 times the length of the shorter leg. Special Right Triangles 45 2 Remember Pythagorean Theorem a 2 + b 2 = c 2 2y 60 y 45 Pythagorean Theorem a 2 + b 2 = c 2 2 + 2 = ( 2) 2 2 2 = 2 2 30 y 3 Pythagorean Theorem a 2 + b 2 = c 2 y 2 + (y 3) 2 = (2y) 2 y 2 + 3y 2 = 4y 2 Eample 1: The wallpaper in the figure can be divided into four equal square quadrants so that each square contains 8 triangles. What is the area of one of the squares if the hypotenuse of each 45-45 -90 triangle measures 7 2 millimeters? Eample 2: If each 45-45 -90 triangle in the figure has a hypotenuse of 5 2 millimeters, what is the perimeter of the entire square?
Eample 3: Find a in RST Eample 4: Find b in ABC Eample 5: Find QR in PQR Eample 6: Find BC in ABC Concept Summary: In a 45-45 -90 triangle, the hypotenuse is 2 times the length of the leg. The measures are,, and 2 In a 30-60 -90 triangle, the measures of the sides are, 3, and 2. Homework: pg 556-57; 1-6, 11-13, 29, 30
opposite adjacent Chapter 8: Right Triangles and Trigonometry Section 8-4: Trigonometry SOL: G.7 The student will solve practical problems involving right triangles by using the Pythagorean Theorem, properties of special right triangles, and right triangle trigonometry. Solutions will be epressed in radical form or as decimal approimations. Objective: Find trigonometric ratios using right triangles Solve problems using trigonometric ratios Vocabulary: Trigonometry (or Trig) involves triangle angular measurement Inverses, (trig function -1 ), finds the angular measurement Key Concept: Trig Common Angles Function Definition 0 30 45 60 90 Sin Opposite over Hypotenuse 0 ½ 2/2 3/2 0.500 0.707 0.866 1 Cos Adjacent over Hypotenuse 1 3/2 2/2 ½ 0.866 0.707 0.500 0 Tan Opposite over Adjacent 0 3/3 3 1 0.577 1.732 SOH-CAH-TOA (Sin = O/H, Cos = A/H, Tan = O/A) Trigonometry B a c Pythagorean Theorem a 2 + b 2 = c 2 a 2 b 2 -- + -- = 1 c 2 c 2 Opposite 2 Adjacent 2 ---------------- + ----------------- = 1 Hypotenuse 2 Hypotenuse 2 C A A sin 2 A + cos 2 A = 1 b Trig Crash Recovery Course A θ hypotenuse hypotenuse C adjacent θ B What s Constant: Side opposite right angle is the hypotenuse What Changes: Side opposite the angle, θ; Side adjacent to the angle, θ In the triangle to the left: AC is opposite of θ and BC is adjacent to it In the triangle to the right: AC is adjacent to θ and BC is opposite it Left-most Triangle: opposite side AC sin θ = ------------------- = ------ hypotenuse AB adjacent side BC cos θ = ------------------- = ------ hypotenuse AB opposite side AC tan θ = ------------------- = ------ adjacent side BC C opposite Right-most Triangle: opposite side BC sin θ = ------------------- = ------ hypotenuse AB adjacent side AC cos θ = ------------------- = ------ hypotenuse AB opposite side BC tan θ = ------------------- = ------ adjacent side AC B
Anatomy of a Trig Function Main Trig Functions: Sin Cos Tan Others: Csc Sec Cot A θ hypotenuse C B Use trig functions to help find a missing side in a right triangle. Format: some side Trig Function ( an angle, θ for eample) = ----------------------- some other side where the some side or the some other side is the missing side (variable, like ) Use inverse trig functions to help find a missing angle in a right triangle. Format: some side Trig Function -1 (-------------------------) = missing angle, θ for eample some other side where the trig function -1 is found using 2 nd key then the trig function on calculator Eamples and Other Things SOH-CAH-TOA (Sin θ = O/H, Cos θ = A/H, Tan θ = O/A) θ is a symbol for an angle sin 2 θ + cos 2 θ = 1 (from Pythagorean Theorem) Remember: Sin 90 is 1 Cos 90 is 0 Tan 90 is undefined 12 8 tan = 12/8 = tan -1 (12/8) = 56.31 When looking for an angle use the inverse of the appropriate trig function (2 nd key then trig function on your calculator) 12 is opposite the angle ; and 8 is adjacent to it: opp, adj use tangent 52 17 tan 52 = 17/ tan 52 = 17 = 17/tan 52 = 13.28 When looking for a side use the appropriate trig function (based on your angle and its relationship to, and your given side). 17 is opposite of the angle and is adjacent to it: opp, adj use tangent 13 sin 47 = /13 13 sin 47 = 9.51 = 13 is the hypotenuse (opposite from the 90 degree angle) and is opposite from given angle: opp, hyp use sin 47
Eample 1: Find sin L, cos L, tan L, sin N, cos N, and tan N. Epress each ratio as a fraction and as a decimal Eample 2: Find sin A, cos A, tan A, sin B, cos B, and tan B. Epress each ratio as a fraction and as a decimal. Eample 3: Use your calculator to find (to the nearest ten thousandth) a. tan 56 = b. cos 90 = c. sin 48 = d. cos 85 = Eample 1: Find the angle Day Two Eample 2: Find the side Eample 3: Find the side
Eample 4: Check Yourself: You have a hypotenuse and an adjacent side Use: Solve: = You have an opposite and adjacent side Use: Solve: y = You have an opposite side and a hypotenuse Use: Solve: z = Eample 5: A fitness trainer sets the incline on a treadmill to 7. The walking surface is 5 feet long. Approimately how many inches did the trainer raise the end of the treadmill from the floor? Eample 6: The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8, how high does the ramp rise off the ground to the nearest inch? Trig Practice Board work 1. 2. 20 3. 24 15 30 60 30 4. 5. 6. 13 49 12 45 12 18 7. 8. 16 12 54 10 Concept Summary: Trigonometric ratios can be used to find measures in right triangles Homework: Day 1: pg 567-69; 1-6, 22-4, 31-3 Day 2: pg 567-69; 28-30, 36-9, 43
Section 8-5: Angle of Elevation and Depression SOL: G.7 The student will solve practical problems involving right triangles by using the Pythagorean Theorem, properties of special right triangles, and right triangle trigonometry. Solutions will be epressed in radical form or as decimal approimations. Objective: Solve problems involving angles of elevation Solve problems involving angles of depression Vocabulary: Angle of elevation: angle between line of sight (looking up) and the horizontal Angle of depression: angle between line of sight (looking down) and the horizontal Angles of Elevation and Depression Top Horizontal Angle of Depression Line of Sight Angle of Elevation Bottom Horizontal Since the two horizontal lines are parallel, by Alternate Interior Angles the angle of depression must be equal to the angle of elevation. The angle of depression is complementary to the angle formed by the tower and the line of sight. Angles of Depression or Elevation Step 1: Draw this triangle to fit problem This is the length of string or distance This is the height (above the ground) or depth angle goes here Step 2: Label sides from angle s view Step 3: Identify trig function to use Step 4: Set up equation Step 5: Solve for variable Use inverse trig functions for an angle Θ This is the distance from the base or along the ground
Eample 1: Before the Mast -- At a point on the ground 50 feet from the foot of the flagpole, the angle of elevation to the top of the pole is 53. Find the height of the flagpole. Eample 2: Job Site -- A 20-foot ladder leans against a wall so that the base of the ladder is 8 feet from the base of the building. What angle does the ladder make with the ground? Eample 3: CIRCUS ACTS -- At the circus, a person in the audience watches the high-wire routine. A 5-foot-6- inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member s line of sight to the acrobat is 27? Eample 4: The angle of elevation from point A to the top of the press bo is 49. If point A is 400 ft from the base of the press bo, how high is the press bo? Eample 5: Find the angle of elevation of the sun when a 12.5 ft post casts a 18 ft shadow? Eample 6: A ladder leaning up against a barn makes an angle of 78 with the ground when the ladder is 5 feet from the barn. How long is the ladder? Eample 7: From the top of the 120 foot fire tower a ranger observes a fire at an angle of depression of 19. How far from the base of the tower is the fire? Eample 8: A hill is 1000 yards long with a vertical drop of 208 yards. Find the angle of depression from the top of the hill to the bottom. Concept Summary: Trigonometry can be used to solve problems related to angles of elevation and depression Homework: Day 1: pg 577-79; 1, 2, 7, 9, 10 Day 2: pg 577-79; 13, 15, 16, 20
Section 8-6: Law of Sines and Law of Cosines SOL: None. Objective: Use the Law of Sines to solve triangles Solve problems by using the Law of Sines Use the Law of Cosines to solve triangles Solve problems by using the Law of Cosines Vocabulary: Solving a triangle means finding the measures of all sides and all angles Key Concept: Law of Sines can be used to find missing parts of triangles that are not right triangles Case 1: You know the measures of two angles and any side of the triangle (AAS or ASA) Case 2: You know the measures of two sides and an angle opposite one of the known sides of the triangle (SSA) Law of Sines A Let ABC be any triangle with a, b and c representing the measures of the sides opposite the angles with measures A, B, and C respectively. Then c b B a C sin A sin B sin C = = a b c Law of Cosines can be used to solve triangles when the Law of Sines cannot be used Case 1: You know the measures of two sides and their included angle (SAS) Case 2: You know the measures of all three sides (SSS)
Law of Cosines A Let ABC be any triangle with a, b and c representing the measures of the sides opposite the angles with measures A, B, and C respectively. Then the following equations are true: B c a b C a 2 = b 2 + c 2 2bc cos A b 2 = a 2 + c 2 2ac cos B c 2 = a 2 + b 2 2ab cos C Concept Summary: To solve a triangle means to find the measures of all sides and angles To find the measures of a triangle by using the Law of Sines, you must either know the measures of two angle and any side (AAS or ASA), or two sides and an angle opposite one of these sides (SSA) of the triangle The Law of Cosines can be used to solve triangles when you know the measures of two sides and the included angle (SAS) or the measures of all three sides (SSS) Homework: pg 586-7; 1-6, 13-16
Section 8-7: Vectors TBD SOL: None. Objective: Perform vector operations geometrically Perform vector operations on the coordinate plane Vocabulary: component form like a point but in <,y> that gives the component and y component of the force vector direction an angle at which the force is applied (-ais is zero) magnitude length of a vector (hypotenuse of its component forms) resultant when two vectors are added together standard position starts at the origin vector a force that describes both magnitude and direction Key Concept:
Concept Summary: Homework: pg 598-9; 5-8, 22-24, 29,31
Decision Tree for Trig Problems Right Triangle Unknown Triangle Special Triangles Use Law of Sines or Law of Cosines 45-45-90 30-60-90 Rule Known Law of Measures of two angles and any side (AAS or ASA) Non-Special Triangles Sines Measures of two sides and an angle opposite one of these sides (SSA) Use regular sin, cos, and tan Law of Cosines Measures of two sides and the included angle (SAS) Measure of all three sides (SSS) Use inverses to find angular measurement We will revisit trig functions when we get into circles (Chapter 10). Trigonometric Tips π/2 90 1 y π 180 A 0 2π 270 3π/2 From Pythagorean Thrm: 2 + y 2 = 1 From Trig: sin 2 A + cos 2 A = 1 In a Unit Circle: sin A = y cos A = For some interesting JAVA applets on trigonometry: sine and cosine: http://www.catcode.com/trig/trig05.html tangent: http://www.catcode.com/trig/trig12.html circles: http://www.catcode.com/trig/trig06.html For some interesting JAVA applets on trigonometry: Sine and cosine: http://www.catcode.com/trig/trig05.html Tangent: http://www.catcode.com/trig/trig12.html Circles: http://www.catcode.com/trig/trig06.html
Lesson 8-1: Find the geometric mean between each pair of numbers. State eact answers and answers to the nearest tenth. 1. 9 and 13 2. 2 5 and 5 5 3. Find the altitude 4. Find, y, and z 5. Which of the following is the best estimate of? Lesson 8-2: Find. 1. 2. 3. Determine whether QRS with vertices Q(2,-3), R(0,-1), and S(4,-1) is a right triangle. If so, identify the right angle. Determine whether each set of measures forms a right triangle and state whether they form a Pythagorean triple. 4. 16, 30, 33 5. 5/8, 3/2, 13/8 6. Which of the following are not the lengths of sides of a right triangle? a. 25, 20, 15 b. 4, 7.5, 8.5 c. 0.7, 2.4, 2.5 d. 36, 48, 62
Lesson 8-3: Find and y. 1. 2. 3. The length of a diagonal of a square is 15 2 cm. Find the perimeter of the square. 4. The side of an equilateral triangle measures 21 inches. Find the length of the altitude of the triangle. 5. MNP is a 45-45 - 90 triangle with right angle P. Find the coordinates of M in quadrant II with P(2,3) and N(2,8). 6. In the right triangle find CD if DE = 5.? Lesson 8-4a: Find. 1. 2. 3. Given an adjacent side and the hypotenuse, which trig function do you use? 4. Given an opposite side and the hypotenuse, which trig function do you use? 5. MNP is a 45-45 - 90 triangle with right angle P. Find NP if MN = 20. 6. In the right triangle which trig function would you use to find CD with C? a. sin b. cos c. tan d. sec Lesson 8-4b: 1. Use a graphing calculator to find tan 54. Round to the nearest ten-thousandth. 2. Find m B to the nearest tenth of a degree if cos B = 0.8926 and B is an acute angle. Find. Round the nearest tenth. 3. 4. 5. 6. What is the value of tan Θ? a. 12/5 b. 12/13 c. 5/12 d. 5/13
Lesson 8-5: Name the angles of depression and elevation in the two figures. 1. 2. 3. Find the angle of elevation of the sun when a 6-meter flag pole casts a 17-meter shadow. 4. After flying at an altitude of 575 meters, a helicopter starts to descend when its ground distance from the landing pad is 13.5 Km. What is the angle of depression for this part of the flight? 5. The top of a signal tower is 250 feet above sea level. The angle of depression for the tope of the tower to a passing ship is 19. How far is the foot of the tower from the ship? 6. From a point 50 feet from the base of a tree, the angle of elevation to the top of the tree is 32. From a point closer to the base of the tree, the angle of elevation is 64. Which of the following is the best estimate of the distance between the two points at which the angle of elevation is measured? a. 15 b. 28 c. 29 d. 35